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We employ an $n$-replica Keldysh field theory to investigate the effects ofmeasurements and decoherence on long distance behaviors of quantum criticalstates. We classify different measurements and decoherence based on theirtimescales and symmetry properties, and demonstrate that they can be describedby $n$-replica Keldysh field theories with distinct physical and replicasymmetries. Low energy effective theories for various scenarios are thenderived using the symmetry and fundamental consistency conditions of theKeldysh formalism. We apply this framework to study the critical Ising model inboth one and two spatial dimensions. In one dimension, we demonstrate that (1)measurements over a finite period of time along the transverse spin directiondo not modify the asymptotic scaling of correlation functions and entanglemententropy, whereas (2) measurements along the longitudinal spin direction lead toan area law entangled phase. We also show that (3) decoherence noises over afinite time can be mapped to specific boundary conditions of a criticalAshkin-Teller model, and the entanglement characteristics of the resultingmixed state can be determined. For measurements and decoherence over anextensive time, we demonstrate that (4) the von Neumann entanglement entropy ofa large subsystem can exhibit a (sub-)dominant logarithmic scaling in thestationary state for weak measurement (decoherence) performed in a basis thatis symmetric under the Ising symmetry, but (5) reduces to an area law formeasurements and decoherence in the longitudinal direction. Our resultsdemonstrate that the Keldysh formalism is a useful tool for systematicallystudying the effects of measurements and decoherence on long-wavelengthphysics.

Exploring critical systems under measurements and decoherence via Keldysh field theory